Going to and fro along the track, the little man would judge that the rails are not equidistant at all. Applying his shrinkable foot-rule, he would decide that the space between the rails varied in width at different places. He would come to the same conclusion that we have set down, namely, that parallels may at first approach but, following them further, they diverge more and more. All this he would discover and never suspect that his own variable dimensions were the cause of.a deception, for would not his surroundings shrink always in proportion to himself? Beltrami's illustration thus attains its purpose by making solid objects expand and contract in place of allowing space itself to grow any more roomy than our Euclidean notions permit.
If Euclid were to return to earth to-day, he would find many geometries, but the three here described would probably interest him above all others. A fitting task for Euclid would be to coordinate this trilogy of systems. With the three volumes spread open before him, he could write a dictionary by the aid of which a student could translate any proposition stated in one volume into the corresponding proposition given in either of the other two. It is at bottom a matter of words, or at least the facts lend themselves to that interpretation. With intense satisfaction Euclid could still contemplate his own geometry. "Where long and involved phrases are necessary to convey the idea presented in the other systems, his own ideas are always lucid and tersely expressible. His system is consequently by far the most convenient, so much more convenient, that if we should ever discover any discrepancy between it and the facts of the physical universe, we would probably prefer to change our laws of physics or mechanics rather than to adjust ourselves to a less convenient system of geometry.